Chapter 10 Circles

Case Study 1: The Park's Fountain

Case Description:
In a circular park, there is a beautiful fountain located at the center. The diameter of the fountain is 4 meters. The park has a path that runs around the fountain, creating a circular area around it with a radius of 10 meters. The management of the park wants to calculate the area of the path that surrounds the fountain to plan for landscaping and seating arrangements.

The area of the path can be calculated by finding the difference between the area of the larger circle (path) and the area of the smaller circle (fountain). The area of a circle is calculated using the formula:

$$\text{Area} = \pi r^2$$

Where $r$ is the radius of the circle.

MCQs:

1. What is the radius of the fountain?

   • A) 2 meters
   • B) 4 meters
   • C) 6 meters
   • D) 10 meters

2. What is the radius of the circular path around the fountain?

   • A) 4 meters
   • B) 8 meters
   • C) 10 meters
   • D) 12 meters

3. Using $\pi \approx 3.14$, what is the area of the fountain?

   • A) 12.56 m²
   • B) 25.12 m²
   • C) 50.24 m²
   • D) 78.5 m²

4. What is the area of the path surrounding the fountain?

   • A) 25.12 m²
   • B) 78.5 m²
   • C) 153.86 m²
   • D) 201.06 m²

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Case Study 2: The Circular Garden

Case Description:
A circular garden has a radius of 5 meters. A gardener plans to plant flowers in the inner circle while leaving a uniform width of 1 meter all around for walking space. To ensure the flowers have enough room, the gardener needs to calculate the area of the inner circle where the flowers will be planted and the area of the walking space.

The area for the flowers can be calculated using the same formula for the area of a circle. The outer radius will be the original radius of the garden, while the inner radius will be the radius minus the width of the walking space.

MCQs:

1. What is the radius of the circular garden?

   • A) 4 meters
   • B) 5 meters
   • C) 6 meters
   • D) 7 meters

2. What is the radius of the inner circle where the flowers will be planted?

   • A) 4 meters
   • B) 5 meters
   • C) 3 meters
   • D) 2 meters

3. What is the area of the circular garden?

   • A) 25π m²
   • B) 20π m²
   • C) 30π m²
   • D) 35π m²

4. What is the area of the walking space?

   • A) 6π m²
   • B) 4π m²
   • C) 5π m²
   • D) 1π m²

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Case Study 3: The Clock Face

Case Description:
A clock has a circular face with a radius of 15 cm. The minute hand of the clock is 10 cm long. The clockmaker wants to calculate the area of the clock face and the area that the minute hand sweeps as it moves from the 12 o'clock position to the 6 o'clock position.

To find the area swept by the minute hand, the clockmaker will treat it as a sector of the circle, where the angle covered from 12 to 6 is 180 degrees.

MCQs:

1. What is the radius of the clock face?

   • A) 10 cm
   • B) 12 cm
   • C) 15 cm
   • D) 20 cm

2. What is the radius of the minute hand?

   • A) 5 cm
   • B) 10 cm
   • C) 15 cm
   • D) 20 cm

3. Using $\pi \approx 3.14$, what is the area of the clock face?

   • A) 100.5 cm²
   • B) 200.5 cm²
   • C) 706.5 cm²
   • D) 150.5 cm²

4. What is the area of the sector swept by the minute hand?

   • A) 25 cm²
   • B) 30 cm²
   • C) 35 cm²
   • D) 78.5 cm²

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Case Study 4: The Bicycle Wheel

Case Description:
A bicycle wheel is circular with a radius of 35 cm. The cyclist wants to find out how far the bicycle will travel in one complete rotation of the wheel. This distance is known as the circumference of the wheel, calculated using the formula:

$$\text{Circumference} = 2\pi r$$

Where $r$ is the radius of the wheel. Knowing the circumference will help the cyclist plan their route and estimate the distance covered.

MCQs:

1. What is the radius of the bicycle wheel?

   • A) 30 cm
   • B) 35 cm
   • C) 40 cm
   • D) 45 cm

2. Using $\pi \approx 3.14$, what is the circumference of the wheel?

   • A) 70 cm
   • B) 150 cm
   • C) 220 cm
   • D) 80 cm

3. If the cyclist makes 5 complete rotations, what will be the total distance traveled?

   • A) 150 cm
   • B) 350 cm
   • C) 500 cm
   • D) 1000 cm

4. What is the distance traveled in kilometers if the cyclist makes 10 rotations?

   • A) 0.5 km
   • B) 1.0 km
   • C) 2.0 km
   • D) 0.2 km

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Case Study 5: The Sports Track

Case Description:
A sports track is circular with a radius of 25 meters. The track has a width of 2 meters. The event organizers want to calculate the area of the track to set up seating and viewing areas for spectators. The total area of the track can be found by calculating the area of the outer circle and subtracting the area of the inner circle.

MCQs:

1. What is the radius of the sports track?

   • A) 20 meters
   • B) 23 meters
   • C) 25 meters
   • D) 30 meters

2. What is the radius of the inner circle of the track?

   • A) 23 meters
   • B) 25 meters
   • C) 27 meters
   • D) 30 meters

3. Using $\pi \approx 3.14$, what is the area of the outer circle?

   • A) 2000 m²
   • B) 1950 m²
   • C) 2500 m²
   • D) 3000 m²

4. What is the area of the track itself?

   • A) 250 m²
   • B) 300 m²
   • C) 400 m²
   • D) 450 m²